Total Scalar Curvature and L 2 Harmonic 1-Forms on a Minimal Hypersurface in Euclidean Space

Let M ⁿ , n 3, be a complete oriented immersed minimal hypersurface in Euclidean space R ⁿ⁺¹. We show that if the total scalar curvature on M is less than the n/2 power of 1/C _s , where C _s is the Sobolev constant for M, then there are no L ² harmonic 1-forms on M. As corollaries, such a minimal hypersurface contains no nontrivial harmonic functions with finite Dirichlet integral and so it has only one end. This implies finally that M is a hyperplane.     

Constant Scalar Curvature Metrics on Toric Surfaces

The main result of the paper is an existence theorem for a constant scalar curvature Kahler metric on a toric surface, assuming the K-stability of the manifold. The proof builds on earlier papers by the author, which reduce the problem to certain a priori estimates. These estimates are obtained using a combination of arguments from Riemannian geometry and convex analysis. The last part of the paper contains a discussion of the phenomena that can be expected when the K-stability does not hold and solutions do not exist. Received: May 2008, Revision: December 2008, Accepted: December 2008     

Quasi—conformally Flat Manifolds with Constant Scalar Curvature

§１．　ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＭｂｅａｎｎ－ｄｉｍｅｎｓｉｏｎａｌｃｏｎｆｏｒｍａｌｌｙｆｌａｔｍａｎｉｆｏｌｄｗｉｔｈｃｏｎｓｔａｎｔｓｃａｌａｒｃｕｒｖａｔｕｒｅρ（ｎ≥３）．ＷｈｅｎｔｈｅＲｉｃｃｉｃｕｒｖａｔｕｒｅＳｏｆＭｉｓｏｆｂｏｕｎｄｅｄｂｅｌｏｗａｎｄｙＳｙ２＜ρ２／（ｎ－１），Ｇｏｌｄ－ｂｅｒｇｐｒｏｖｅｄｔｈａｔＭｉｓｏｆｃｏｎｓｔａｎｔｃｕｒｖａｔｕｒｅ［１］．ＷｈｅｎＭｉｓａｃｏｍｐａｃｔｍａｎｉｆｏｌｄｗｉｔｈｐｏｓｉｔｉｖｅＲｉｃｃｉｃｕｒｖａｔｕｒｅ，ＷｕＢ…     

空间形式中常数量率的子流形

本文利用一个类似于Cheng和Yau引进的微分算子的新微分算子，得到了单位球面中常数量率的紧致子流形的一个刚性结果．     

Unit Tangent Sphere Bundles with Constant Scalar Curvature

As a first step in the search for curvature homogeneous unit tangent sphere bundles we derive necessary and sufficient conditions for a manifold to have a unit tangent sphere bundle with constant scalar curvature. We give complete classifications for low dimensions and for conformally flat manifolds. Further, we determine when the unit tangent sphere bundle is Einstein or Ricci-parallel.     

Complete Hypersurfaces with Constant Laguerre Scalar Curvature in R~n

Let x:M~(n-1)→R~n be an umbilical free hypersurface with non-zero principal curvatures.Two basic invariants of M under the Laguerre transformation group of R~n are Laguerre form C and Laguerre tensor L.In this paper,we prove the following theorem:Let M be an(n-1)-dimensional(n 3) complete hypersurface with vanishing Laguerre form and with constant Laguerre scalar curvature R in R~n,denote the trace-free Laguerre tensor by L =L-1/n-1tr(L)·Id.If sup_M ||L||=0,then M is Laguerre equivalent to a Laguerre isotropic hypersurface;and if sup_M ||L||=((n-1)(n-2))~(1/2)R/((n-1)(n-2)(n-3)),M is Laguerre equivalent to the hypersurface x:H~1×S~(n-2)→R~n.     

关于具有常平均曲率和数量曲率超曲面的Moebius几何的一个注记

设x：M→S^n＋1（n≥23）是n＋1-维单位球中的无脐点超曲面，Moebius不变量G，Ф，A和B分别表示x的Moebius度量，Moebius形式，Blaschke形式和Moebius第二基本形式．本文证明了如果x的Moebius形式圣平行，并且A＋λG＋μB=0，其中λ，μ分别是定义在M上的光滑函数，那么Ф=0，由此及李海中、王长平（2003年）文献中的分类定理给出了眇州中具有平行的Moebius形式及满足A＋λG＋μB=0的超曲面的分类．此结果推广了他们及张廷枋（2003年）文献中的结果．     

A Characterization of Quadric Constant Scalar Curvature Hypersurfaces of Spheres

Let M?S ⁴ be a complete orientable hypersurface with constant scalar curvature. For any v∈R ⁵, let us define the two real functions \(l_{v}, f_{v}:M\to{\bf R}\) on M by l _v (x)=〈x,v〉 and f _v (x)=〈ν(x),v〉 with ν:M→S ⁴ a Gauss map of M. In this paper, we show that if we have that l _v =λf _v for some nonzero vector v∈R ⁵ and some real number λ, then M is either totally umbilical (a Euclidean sphere) or M is a Cartesian product of Euclidean spheres. We will also show with an example that the completeness condition is needed in this theorem.     

双曲空间Hn+p(-1)中具常数量曲率的完备子流形

设Mn是Hn p(-1)中具有常标准数量曲率的n维完备子流形,本文证明了这种完备子流形的某些内蕴刚性定理和分类定理,并对超曲面的情形进行了研究．     

A Construction of Constant Scalar Curvature Manifolds with Delaunay-type Ends

It has been showed by Byde (Indiana Univ. Math. J. 52(5):1147–1199, 2003) that it is possible to attach a Delaunay-type end to a compact nondegenerate manifold of positive constant scalar curvature, provided it is locally conformally flat in a neighborhood of the attaching point. The resulting manifold is noncompact with the same constant scalar curvature. The main goal of this paper is to generalize this result. We will construct a one-parameter family of solutions to the positive singular Yamabe problem for any compact non-degenerate manifold with Weyl tensor vanishing to sufficiently high order at the singular point. If the dimension is at most 5, no condition on the Weyl tensor is needed. We will use perturbation techniques and gluing methods.     

复射影空间中具有常数量曲率的实超曲面

将球面上常数量曲率超曲面推广到复射影空间中,得到此类实超曲面的某些积分不等式.     

Minimal Submanifolds in Sp+3 with Constant Scalar Curvature

Let M be a compact, minimal 3-dimensional submanifold with constant scalar curvature R immersed in the standard sphere S^3+p. In codimension 1, we know from the work that has been done on Chern’s conjecture that M is isoparametric and R = 3D0, R = 3D3 or R = 3D6. In this paper we extend this result from codimension one to compact submanifolds with a flat normal bundle and give a complete classification.     

Rigidity Theorems for Complete Sasakian Manifolds with Constant Pseudo-Hermitian Scalar Curvature

The orthogonal decomposition of the Webster curvature provides us a way to characterize some canonical metrics on a pseudo-Hermitian manifold. We derive some subelliptic differential inequalities from the Weitzenböck formulas for the traceless pseudo-Hermitian Ricci tensor of Sasakian manifolds with constant pseudo-Hermitian scalar curvature and the Chern–Moser tensor of the Sasakian pseudo-Einstein manifolds, respectively. By means of either subelliptic estimates or maximum principle, some rigidity theorems are established to characterize Sasakian pseudo-Einstein manifolds among Sasakian manifolds with constant pseudo-Hermitian scalar curvature and Sasakian space forms among Sasakian pseudo-Einstein manifolds, respectively.     

L 2-Harmonic 1-Forms on Submanifolds with Finite Total Curvature

Let $x:M^{m}\to\bar{M}$ , m≥3, be an isometric immersion of a complete noncompact manifold M in a complete simply connected manifold $\bar{M}$ with sectional curvature satisfying $-k^{2}\leq K_{\bar{M}}\leq0$ , for some constant k. Assume that the immersion has finite total curvature in the sense that the traceless second fundamental form has finite L ^m -norm. If $K_{\bar{M}}\not\equiv0$ , assume further that the first eigenvalue of the Laplacian of M is bounded from below by a suitable constant. We prove that the space of the L ² harmonic 1-forms on M has finite dimension. Moreover, there exists a constant Λ>0, explicitly computed, such that if the total curvature is bounded from above by Λ then there are no nontrivial L ²-harmonic 1-forms on M.     

空间形式中具常数量曲率的子流形的内蕴刚性

Under the assumption that the normalized mean curvature vector is parallel in the normal bundle, by using the generalized ChengYau＇s self-adjoint differential operator, here we obtain some rigidity results for compact submanifolds with constant scalar curvature and higher codimension in the space forms.     

Rigidity Theorem of Hypersurfaces with Constant Scalar Curvature in a Unit Sphere

In this paper, we give a characterization of tori S^1 （ √ nr＋2-n/nr）×S^n-1（√ n-2/nr） and S^m （ √n/m ） ×S^n-m （√n-m/n）. Our result extends the result due to Li （1996）on the condition that M is an n-dimensional complete hypersurface in Sn＋1 with two distinct principal curvatures. Keywords principal curvature, Clifford torus, Gauss equations     

局部对称Lorentz空间中的具有常数量曲率的完备类空超曲面

The complete space-like hypersurfaces with constant normal saclar curvature is discussed in a locally symmetric Lorentz space. A classified theorem is obtained by the operator L_1 introduced by S Y Cheng and S T Yau [3].